A174096 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=2, read by rows.

1, 1, 1, 1, 12, 1, 1, 12, 12, 1, 1, 12, 16, 12, 1, 1, 13, 17, 17, 13, 1, 1, 16, 36, 32, 36, 16, 1, 1, 17, 49, 37, 37, 49, 17, 1, 1, 32, 93, 92, 36, 92, 93, 32, 1, 1, 33, 124, 197, 80, 80, 197, 124, 33, 1, 1, 36, 204, 304, 197, 44, 197, 304, 204, 36, 1

Offset

0,5

Comments

Row sums are: 1, 2, 14, 26, 42, 62, 138, 208, 472, 870, 1528, ...

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 2.

Example

Triangle begins as:
  1;
  1,  1;
  1, 12,   1;
  1, 12,  12,   1;
  1, 12,  16,  12,   1;
  1, 13,  17,  17,  13,  1;
  1, 16,  36,  32,  36, 16,   1;
  1, 17,  49,  37,  37, 49,  17,   1;
  1, 32,  93,  92,  36, 92,  93,  32,   1;
  1, 33, 124, 197,  80, 80, 197, 124,  33,  1;
  1, 36, 204, 304, 197, 44, 197, 304, 204, 36, 1;

Mathematica

A174093[n_, k_]:= If[n<2, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
T[n_, k_, q_]:= Sum[q^j*Floor[A174093[n, k]/2^j], {j, 0, 10}];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 10 2021 *)

Prog

(Sage)
def A174093(n, k): return 1 if n<2 else binomial(n-k+1, k) + binomial(k+1, n-k)
def T(n, k, q): return sum( q^j*(A174093(n, k)//2^j) for j in (0..10) )
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021
(Magma)
A174093:= func< n, k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
T:= func< n, k, q | (&+[ q^j*Floor(A174093(n, k)/2^j): j in [0..10]]) >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Crossrefs

Cf. A174093 (q=0), A174095 (q=1), this sequence (q=2), A174097 (q=3).

Sequence in context: A058306 A010207 A010206 * A070636 A168646 A051457

Adjacent sequences: A174093 A174094 A174095 * A174097 A174098 A174099

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Mar 07 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved